Duke Mathematical Journal

Serrin’s overdetermined problem and constant mean curvature surfaces

Manuel Del Pino, Frank Pacard, and Juncheng Wei

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Abstract

For all N9, we find smooth entire epigraphs in RN, namely, smooth domains of the form Ω:={xRN|xN>F(x1,,xN1)}, which are not half-spaces and in which a problem of the form Δu+f(u)=0 in Ω has a positive, bounded solution with 0 Dirichlet boundary data and constant Neumann boundary data on Ω. This answers negatively for large dimensions a question by Berestycki, Caffarelli, and Nirenberg. In 1971, Serrin proved that a bounded domain where such an overdetermined problem is solvable must be a ball, in analogy to a famous result by Alexandrov that states that an embedded compact surface with constant mean curvature (CMC) in Euclidean space must be a sphere. In lower dimensions we succeed in providing examples for domains whose boundary is close to large dilations of a given CMC surface where Serrin’s overdetermined problem is solvable.

Article information

Source
Duke Math. J., Volume 164, Number 14 (2015), 2643-2722.

Dates
Received: 15 October 2013
Revised: 9 November 2014
First available in Project Euclid: 26 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1445865570

Digital Object Identifier
doi:10.1215/00127094-3146710

Mathematical Reviews number (MathSciNet)
MR3417183

Zentralblatt MATH identifier
1342.35188

Subjects
Primary: 35J25: Boundary value problems for second-order elliptic equations
Secondary: 35J67: Boundary values of solutions to elliptic equations

Keywords
overdetermined elliptic equation constant mean curvature surface entire minimal graph

Citation

Del Pino, Manuel; Pacard, Frank; Wei, Juncheng. Serrin’s overdetermined problem and constant mean curvature surfaces. Duke Math. J. 164 (2015), no. 14, 2643--2722. doi:10.1215/00127094-3146710. https://projecteuclid.org/euclid.dmj/1445865570


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References

  • [1] A. D. Alexandrov. Uniqueness theorems for surfaces in the large, I (in Russian), Vestnik Leningrad Univ. 11 (1956), 5–17.
  • [2] L. Ambrosio and X. Cabré, Entire solutions of semilinear elliptic equations in $\mathbb{R} ^{3}$ and a conjecture of De Giorgi, J. Amer. Math. Soc. 13 (2000), 725–739.
  • [3] J. L. Barbosa, M. do Carmo, and J. Eschenburg, Stability of hypersurfaces of constant mean curvature in Riemannian manifolds, Math. Z. 197 (1988), 123–138.
  • [4] H. Berestycki, L. A. Caffarelli, and L. Nirenberg, Monotonicity for elliptic equations in unbounded Lipschitz domains, Comm. Pure Appl. Math. 50 (1997), 1089–1111.
  • [5] E. Bombieri, E. De Giorgi, and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math. 7 (1969), 243–268.
  • [6] C. J. Costa, Example of a complete minimal immersions in $\mathbb{R} ^{3}$ of genus one and three embedded ends, Bol. Soc. Brasil. Mat. 15 (1984), 47–54.
  • [7] E. De Giorgi, “Convergence problems for functionals and operators” in Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978), Pitagora, Bologna, 1979, 131–188.
  • [8] M. del Pino, M. Kowalczyk, and J. Wei, On De Giorgi’s conjecture in dimension $N\ge9$, Ann. of Math. (2) 174 (2011), 1485–1569.
  • [9] M. del Pino, M. Kowalczyk, and J. Wei, Entire solutions of the Allen-Cahn equation and complete embedded minimal surfaces of finite total curvature in $\mathbb{R} ^{3}$, J. Differential Geom. 93 (2013), 67–131.
  • [10] A. Farina, L. Mari, and E. Valdinoci, Splitting theorems, symmetry results and overdetermined problems for Riemannian manifolds, Comm. Partial Differential Equations 38 (2013), 1818–1862.
  • [11] A. Farina and E. Valdinoci, Flattening results for elliptic PDEs in unbounded domains with applications to overdetermined problems, Arch. Ration. Mech. Anal. 195 (2010), 1025–1058.
  • [12] A. Farina and E. Valdinoci, 1D symmetry for solutions of semilinear and quasilinear elliptic equations, Trans. Amer. Math. Soc. 363, no. 2 (2011), 579–609.
  • [13] N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems, Math. Ann. 311 (1998), 481–491.
  • [14] B. Gidas, W. M. Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209–243.
  • [15] L. Hauswirth, F. Hélein, and F. Pacard, On an overdetermined elliptic problem, Pacific J. Math. 250 (2011), 319–334.
  • [16] D. Hoffman and W. H. Meeks III, Embedded minimal surfaces of finite topology, Ann. of Math. (2) 131 (1990), 1–34.
  • [17] R. Mazzeo and F. Pacard, Constant mean curvature surfaces with Delaunay ends, Comm. Anal. Geom. 9 (2001), 169–237.
  • [18] F. Morabito, Index and nullity of the Gauss map of the Costa-Hoffman-Meeks surfaces, Indiana Univ. Math. J. 58 (2009), 677–707.
  • [19] S. Nayatani, Morse index and Gauss maps of complete minimal surfaces in Euclidean $3$-space, Comment. Math. Helv. 68 (1993), 511–537.
  • [20] F. Pacard and M. Ritoré, From constant mean curvature hypersurfaces to the gradient theory of phase transitions, J. Differential Geom. 64 (2003), 359–423.
  • [21] A. Ros and P. Sicbaldi, Geometry and topology of some overdetermined elliptic problems, J. Differential Equations 255 (2013), 951–977.
  • [22] O. Savin, Regularity of flat level sets in phase transitions, Ann. of Math. (2) 169 (2009), 41–78.
  • [23] F. Schlenk and P. Sicbaldi, Bifurcating extremal domains for the first eigenvalue of the Laplacian, Adv. Math. 229 (2012), 602–632.
  • [24] J. Serrin, A symmetry problem in potential theory, Arch. Ration. Mech. Anal. 43 (1971), 304–318.
  • [25] P. Sicbaldi, New extremal domains for the first eigenvalue of the Laplacian in flat tori, Calc. Var. Partial Differential Equations 37 (2010), 329–344.
  • [26] J. Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105.
  • [27] M. Traizet, Classification of the solutions to an overdetermined elliptic problem in the plane, Geom. Funct. Anal. 24 (2014), 690–720.
  • [28] B. White, The space of minimal submanifolds for varying Riemannian metrics, Indiana Univ. Math. J. 40 (1991), 161–200.